The garage opener was installed a couple feet away from the door which makes it awkward to trigger in enough situations that I finally just (temporarily) nailed it over next to the doorway. But I need to do something more permanent and less ugly.

The problem is what to do with the wire? There are two studs between where it's mounted and the door.

I'm thinking that since I have to repair the original hole anyway, maybe enlarge it to expose the stud, chisel or drill a path for the wire through it, and pop it up there above the light switch box.

Anyone have better ideas? And suggestions on how to patch the hole?

[Update]

I opened up the wall around the hole to expose the 2x4. As a complication, there's a wood backboard behind the dryway, so had to drill/chisel through that. I didn't have a coat hanger, but had a similar stiffness wire that I used to poke through the insulation. I twisted the end of the stiff wire into a loop using needlenose pliers and attached a flexible wire, pulled that through, and tied it around the door opener wire. With all that done, the garage opener wire was easily pulled through the hole and reconnected to the opener.